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Theorem mrsubval 30660
 Description: The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mrsubffval.c 𝐶 = (mCN‘𝑇)
mrsubffval.v 𝑉 = (mVR‘𝑇)
mrsubffval.r 𝑅 = (mREx‘𝑇)
mrsubffval.s 𝑆 = (mRSubst‘𝑇)
mrsubffval.g 𝐺 = (freeMnd‘(𝐶𝑉))
Assertion
Ref Expression
mrsubval ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → ((𝑆𝐹)‘𝑋) = (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
Distinct variable groups:   𝑣,𝐴   𝑣,𝐶   𝑣,𝐹   𝑣,𝑅   𝑣,𝑋   𝑣,𝑇   𝑣,𝑉
Allowed substitution hints:   𝑆(𝑣)   𝐺(𝑣)

Proof of Theorem mrsubval
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 mrsubffval.c . . . 4 𝐶 = (mCN‘𝑇)
2 mrsubffval.v . . . 4 𝑉 = (mVR‘𝑇)
3 mrsubffval.r . . . 4 𝑅 = (mREx‘𝑇)
4 mrsubffval.s . . . 4 𝑆 = (mRSubst‘𝑇)
5 mrsubffval.g . . . 4 𝐺 = (freeMnd‘(𝐶𝑉))
61, 2, 3, 4, 5mrsubfval 30659 . . 3 ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
763adant3 1074 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → (𝑆𝐹) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
8 simpr 476 . . . 4 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
98coeq2d 5206 . . 3 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) ∧ 𝑒 = 𝑋) → ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) = ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋))
109oveq2d 6565 . 2 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) ∧ 𝑒 = 𝑋) → (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) = (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
11 simp3 1056 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → 𝑋𝑅)
12 ovex 6577 . . 3 (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ∈ V
1312a1i 11 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ∈ V)
147, 10, 11, 13fvmptd 6197 1 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → ((𝑆𝐹)‘𝑋) = (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ifcif 4036   ↦ cmpt 4643   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ⟨“cs1 13149   Σg cgsu 15924  freeMndcfrmd 17207  mCNcmcn 30611  mVRcmvar 30612  mRExcmrex 30617  mRSubstcmrsub 30621 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pm 7747  df-mrsub 30641 This theorem is referenced by:  mrsubcv  30661  mrsub0  30667  mrsubccat  30669
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